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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p><dfn class="terminology">Solution</dfn> The characteristic equation is</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
r_1=\frac{2+\sqrt{4-4 \cdot 5}}{2}=1+2 i,\quad r_2=1-2 i,\quad \lambda=1, \quad \mu=2.
\end{equation*}
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<p class="continuation">The general solution is</p>
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\begin{equation*}
y=C_1 e^{\lambda x} \cos \mu x+C_2 e^{\lambda x} \sin \mu x=C_1 e^{x} \cos (2x)+C_2 e^{x} \sin (2x).
\end{equation*}
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<p class="continuation">Taking the initial conditions into the general solution, one has</p>
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\begin{equation*}
\begin{aligned}
&amp; y\left(\frac{\pi}{2}\right)=0~\to~0=C_1 e^{\frac{\pi}{2}} (-1)~\to~C_1=0~\to~y=C_2 e^{x} \sin (2x),\\
&amp;y^{\prime}\left( \frac{\pi}{2} \right)=2~\to~2=2 C_2 e^{\frac{\pi}{2}} (-1)~\to~C_2=-e^{-\frac{\pi}{2}}.
\end{aligned}
\end{equation*}
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<p class="continuation">Therefore, the solution to the initial value problem is</p>
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\begin{equation*}
y=-e^{-\frac{\pi}{2}} e^{x} \sin (2x).
\end{equation*}
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